Advanced Calculus fi..

Chapter 5 Vector Integral Calculus 371This shows that Ai, ,,,in, is indeed a covariant tensor of order m. As in Section 3.11,it is alternating in all allowed coordinate systems. [We note that the fact that theintegral of an m-form does not change its value under a change of coordinates, aspointed out in Section 5.21, also implies that the m-form is an invariant.]Thus an m-form is an invariant obtained by contraction of a mixed tensor1* $4and summing as in Section 3.9). The result is aninvariant, just as ds2 = g,j dx' dxj is an invariant obtained by contraction.One verifies that the exterior product of alternating covariant tensors, as definedin Section 3.11 , is equivalent to the exterior product of differential forms. (SeeProblem 8 following Section 5.22).Finally formation of the exterior derivative is a valid tensor operation: that is,validity of the relation B = da in one allowed coordinate system implies its validityin all allowed coordinate systems. (See Problem 10 following Section 5.22).(setting jl = il, . . . , j,,, = i!,*5.22 TENSORS AND DIFFERENTIAL FORMSWITHOUT COORDINATESThe elaborate formalism of tensors and differential forms is due to their expressionsin different coordinate systems. It is natural to seek ways of describing the objectsand operations in so far as possible without coordinates. An example is provided bythe interpretation of the gradient of a function F in Section 2.14 as a vector whosedirection is that in which F increases most rapidly and whose length equals that rateof increase.Throughout, all functions will be assumed differentiable as needed.We begin by considering vectors in space, considered here as bound vectors asin a vector field. With each vector v at a point P we can associate the operator v . V(Problem 12 following Section 3.6). When this operator is applied to a function f ,the result is a scalar v . grad f . (If v is a unit vector, then this equals the directionalderivative off at P in the direction of v, but we do not assume that v is a unit vector.)We write L for the differential operator v . V and verify that L has the two properties: .I. L(CI f~ + c2fd = CI Lfl + c2Lf2 (linearity),11. L( f, f2) = f 1 L f2 + f2L fl (product rule).(In I, c,, c2 are arbitrary scalars.) We now de$ne the vectors at P to be the setof all linear operators L satisfying the rules I and I1 at P for all functions fl, f2differentiable in a neighborhood of P. One shows that each such operator L must berepresentable as v . V for a unique v (so that there is a one-to-one correspondencebetween the old vectors v and operators L).The definition applies to n-dimensional space. The set of all vectors at a point P(by the new definition) is called the tangent space at P. The term is suggested by thefact that we can apply the same idea to vectors tangent to a curve, surface, or hypersurfacesat a point P (cf. the "directional derivative along a curve" in Section 2.14).The operators L satisfying I and I1 at P for all functions defined and differentiable-,